2 possible causes... condensation from the evaporator coil inside the air handler is not all going into the drain pan but finding its way onto the air handler case and running to the bottom or...

"but the trap is always cold and also sweating, along with the first foot or so of the drain pipe...." You need to wrap the air handler in insulation because of high ambient air humidity. Hot humid climates require cold surfaces, including ducting, be insulated. You might also try and insulate the PVC drain line.

From this statement, "...The case stays wet on the bottom, with some dripping into the pan beneath," it seems you have 2 drain pans... one in the air handler catching the condensation from the evaporator coil and another outside the air handler to catch and excess water. Usually that second pan is an emergency pan in case the interior pan drain line becomes clogged. There shouldn't be any water in it.